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First order necessary optimality conditions for mathematical programs with second-order cone complementarity constraints

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Abstract

This paper is to develop first order necessary optimality conditions for a mathematical program with second-order cone complementarity constraints (MPSCC) which includes the mathematical program with (vector) complementarity constraints (MPCC) as a special case. Like the case of MPCC, Robinson’s constraint qualification fails at every feasible point of MPSCC if we treat the MPSCC as an ordinary optimization problem. Using the formulas of regular and limiting coderivatives and generalized Clarke’s Jacobian of the projection operator onto second-order cones from the literature, we present the S-, M-, C- and A-stationary conditions for a MPSCC problem. Moreover, several constraint qualifications including MPSCC-Abadie CQ, MPSCC-LICQ, MPSCC-MFCQ and MPSCC-GMFCQ are proposed, under which a local minimizer of MPSCC is shown to be a S-, M-, C- or A-stationary point.

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Acknowledgments

The authors are grateful to the two referees for their valuable comments and suggestions on improving the quality of this paper.

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Authors and Affiliations

  1. Department of Mathematics, School of Science, East China University of Science and Technology, Shanghai, 200237, China

    Yi Zhang

  2. Institute of ORCT, School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    Jia Wu & Liwei Zhang

Authors
  1. Yi Zhang
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  2. Jia Wu
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  3. Liwei Zhang
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Corresponding author

Correspondence to Yi Zhang.

Additional information

This work was supported by the National Natural Science Foundation of China under Projects No. 11401210, No. 11301049, No. 91130007, No. 91330206 and the Fundamental Research Funds for the Central Universities under Project No. 222201314037.

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Zhang, Y., Wu, J. & Zhang, L. First order necessary optimality conditions for mathematical programs with second-order cone complementarity constraints. J Glob Optim 63, 253–279 (2015). https://doi.org/10.1007/s10898-015-0295-2

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  • DOI: https://doi.org/10.1007/s10898-015-0295-2

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